Points on an elliptical orbit where V(radia) is zero.

[Ali Baig]

Homework Statement

Points on an elliptical orbit where the speed is equal to that on a circular orbit?

Homework Equations

The Attempt at a Solution

I have attempted this question and my calculations show that at points on minor and major axes, the radial component of velocity is zero. Hence at these points, the velocity of an orbiter in an elliptical orbit will be equal to that on a circular orbit. Is it correct?

In circular orbit, velocity is always along theta direction and r component of velocity is zero. r-component of velocity is simply time derivative of magnitude of r. So setting dr/dt equal to zero and solving the resulting equation, we can find points at which Vr is zero. Please see the attached picture.

I placed origin of the coordinate system at the "centre" of the ellipse. What will happen if I move the centre of coordinate system to focus of ellipse? Will the results change?

 

[gneill]

Points on an elliptical orbit where the speed is equal to that on a circular orbit?

I'm not sure how to interpret this question. If it assumes a Kepler type gravitational system then the "center" of rotation would be located at one of the foci of the ellipse rather than the ellipse center, and the position of the body in orbit with respect to time will not follow your equation. Also, how are we to decide what the speed for a circular orbit should be? A circular orbit with what radius?

If we use your definition for the elliptical orbit, namely

$\vec{r}(t) = a~cos(t) \hat{i} + b~sin(t) \hat{j}$

then it has an orbital period of $2 \pi$, so presumably the circular orbit is meant to have the same period? But for an arbitrary mathematical orbit like this, not constrained by any particular physical system, the orbit radius can also be arbitrary and so the speed would be arbitrary, too.

Have you stated the problem in full exactly how it was given to you?

 

[mjc123]

There are several problems with this. First, circular orbit of what radius? Semi-major axis? Semi-minor axis? Average of the two? Same radius as the point on the ellipse? Circular orbit with same period?
Second, the question, if correctly transcribed, asks about the orbital speed, not velocity, so direction is not important.
Third, a body in an elliptical orbit orbits about a focus of the ellipse, not the centre. AFAIK (I'm open to correction) an elliptical orbit about the centre is not stable, so there is no meaningful solution to the velocity at a particular point on this "orbit". Even if there was, the points on the axes wouldn't have the same velocity as a circular orbit of the same radius. Consider the point on the minor axis. It is true that Vr = 0, but Vt is not the same as for a circular orbit. It is greater, because it doesn't follow the circular orbit, but a path that takes it further from the centre.